How to join two cylinders with different inside diameters together as one?
For this to work You have to Apply the Solidify modifier and then join the objects. Follow the steps below to accomplish this...


This is almost the copy of Your scene...





Now both objects has the Solidify modifier applied so they have a thickness as could be seen in edit mode here.





Next step is to join both cylinders. To do so select one and hold Shift when selecting the other - press Ctrl + J to finish the process as shown below.





Now we have to delete unnecessary faces so select the part shown below and hit "X" selecting "Only Faces" option to delete them. Next image has this parts already deleted.




Next part requires Loop Tools enabled so enable it in the Preferences as shown below (optionally Save User Setting to keep it enabled).




Now select both inner Loops shown below by using the Alt key. Hold that key and click on one loop (between two vertices) to select the whole circle and then hold Shift + Alt and select the other one to keep both selected like shown below...





Now join them using the Bridge function (red rectangle) as show here...





Select both outer loop and join them too using Bridge again like below...





Switch to object mode and YOU ARE DONE MATE - CONGRATS! Just the final image of the object here...




Hope it helps You, Buddy.